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In the chapter, Approximation Algorithms of Introduction to Algorithm, 3rd Edition, for the approximation problem Travelling Salesman Problem, the author proposes a approximation method that first constructs a minimum spanning tree.

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In order to prove this algorithm is a 2-approximation algorithm, the author claims that:

The weight of the minimum spanning tree $T$ is less than the cost of the optimal tour.

I am wondering if the minimum spanning tree(which is acyclic) of $G$ ensures that its weight is necessarily smaller than any tour(which is cyclic) of the same graph $G$

PS:

The original claim is:

enter image description here

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If you take any tour and remove any edge, then you get a spanning tree. Hence the weight of a minimum spanning tree is at most the weight of every tour.

I should add that one of the tricks behind this algorithm is that while we can't find a minimum tour efficiently, we can find a minimum spanning tree. Minimum spanning tree is an efficiently computable relaxation of minimum Hamiltonian path.

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